Find the Inverse f(x)=3((x-9)/2)+20

$f (x) = 3 (\frac{x - 9}{2}) + 20$

Step 1

Write $f (x) = 3 (\frac{x - 9}{2}) + 20$ as an equation.

$y = 3 (\frac{x - 9}{2}) + 20$

Step 2

Interchange the variables.

$x = 3 (\frac{y - 9}{2}) + 20$

Step 3

Solve for $y$ .

Rewrite the equation as $3 (\frac{y - 9}{2}) + 20 = x$ .

$3 (\frac{y - 9}{2}) + 20 = x$

Simplify $3 (\frac{y - 9}{2}) + 20$ .

Combine $3$ and $\frac{y - 9}{2}$ .

$\frac{3 (y - 9)}{2} + 20 = x$

To write $20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (y - 9)}{2} + 20 \cdot \frac{2}{2} = x$

Simplify terms.

Combine $20$ and $\frac{2}{2}$ .

$\frac{3 (y - 9)}{2} + \frac{20 \cdot 2}{2} = x$

Combine the numerators over the common denominator.

$\frac{3 (y - 9) + 20 \cdot 2}{2} = x$

Simplify the numerator.

Apply the distributive property.

$\frac{3 y + 3 \cdot - 9 + 20 \cdot 2}{2} = x$

Multiply $3$ by $- 9$ .

$\frac{3 y - 27 + 20 \cdot 2}{2} = x$

Multiply $20$ by $2$ .

$\frac{3 y - 27 + 40}{2} = x$

Add $- 27$ and $40$ .

$\frac{3 y + 13}{2} = x$

Multiply both sides by $2$ .

$\frac{3 y + 13}{2} \cdot 2 = x \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 y + 13}{2} \cdot 2 = x \cdot 2$

Rewrite the expression.

$3 y + 13 = x \cdot 2$

Simplify the right side.

Move $2$ to the left of $x$ .

$3 y + 13 = 2 x$

Solve for $y$ .

Subtract $13$ from both sides of the equation.

$3 y = 2 x - 13$

Divide each term in $3 y = 2 x - 13$ by $3$ and simplify.

Divide each term in $3 y = 2 x - 13$ by $3$ .

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{- 13}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{- 13}{3}$

Divide $y$ by $1$ .

$y = \frac{2 x}{3} + \frac{- 13}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{2 x}{3} - \frac{13}{3}$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{2 x}{3} - \frac{13}{3}$

Step 5

Verify if $f^{- 1} (x) = \frac{2 x}{3} - \frac{13}{3}$ is the inverse of $f (x) = 3 (\frac{x - 9}{2}) + 20$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (3 (\frac{x - 9}{2}) + 20)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (3 (\frac{x - 9}{2}) + 20)}{3} - \frac{13}{3}$

Combine the numerators over the common denominator.

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (3 (\frac{x - 9}{2}) + 20) - 13}{3}$

Simplify each term.

Combine $3$ and $\frac{x - 9}{2}$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (\frac{3 (x - 9)}{2} + 20) - 13}{3}$

To write $20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (\frac{3 (x - 9)}{2} + 20 \cdot \frac{2}{2}) - 13}{3}$

Combine $20$ and $\frac{2}{2}$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (\frac{3 (x - 9)}{2} + \frac{20 \cdot 2}{2}) - 13}{3}$

Combine the numerators over the common denominator.

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (\frac{3 (x - 9) + 20 \cdot 2}{2}) - 13}{3}$

Simplify the numerator.

Apply the distributive property.

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (\frac{3 x + 3 \cdot - 9 + 20 \cdot 2}{2}) - 13}{3}$

Multiply $3$ by $- 9$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (\frac{3 x - 27 + 20 \cdot 2}{2}) - 13}{3}$

Multiply $20$ by $2$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (\frac{3 x - 27 + 40}{2}) - 13}{3}$

Add $- 27$ and $40$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (\frac{3 x + 13}{2}) - 13}{3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{2 (\frac{3 x + 13}{2}) - 13}{3}$

Rewrite the expression.

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{3 x + 13 - 13}{3}$

Simplify terms.

Combine the opposite terms in $3 x + 13 - 13$ .

Subtract $13$ from $13$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{3 x + 0}{3}$

Add $3 x$ and $0$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{3 x}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = \frac{3 x}{3}$

Divide $x$ by $1$ .

$f^{- 1} (3 (\frac{x - 9}{2}) + 20) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (\frac{2 x}{3} - \frac{13}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{(\frac{2 x}{3} - \frac{13}{3}) - 9}{2}) + 20$

Simplify each term.

Simplify the numerator.

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{\frac{2 x}{3} - \frac{13}{3} - 9 \cdot \frac{3}{3}}{2}) + 20$

Combine $- 9$ and $\frac{3}{3}$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{\frac{2 x}{3} - \frac{13}{3} + \frac{- 9 \cdot 3}{3}}{2}) + 20$

Combine the numerators over the common denominator.

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{\frac{2 x}{3} + \frac{- 13 - 9 \cdot 3}{3}}{2}) + 20$

Simplify the numerator.

Multiply $- 9$ by $3$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{\frac{2 x}{3} + \frac{- 13 - 27}{3}}{2}) + 20$

Subtract $27$ from $- 13$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{\frac{2 x}{3} + \frac{- 40}{3}}{2}) + 20$

Combine the numerators over the common denominator.

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{\frac{2 x - 40}{3}}{2}) + 20$

Factor $2$ out of $2 x - 40$ .

Factor $2$ out of $2 x$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{\frac{2 (x) - 40}{3}}{2}) + 20$

Factor $2$ out of $- 40$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{\frac{2 x + 2 \cdot - 20}{3}}{2}) + 20$

Factor $2$ out of $2 x + 2 \cdot - 20$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{\frac{2 (x - 20)}{3}}{2}) + 20$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{2 (x - 20)}{3} \cdot \frac{1}{2}) + 20$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{2 (x - 20)}{3} \cdot \frac{1}{2}) + 20$

Rewrite the expression.

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{x - 20}{3}) + 20$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{2 x}{3} - \frac{13}{3}) = 3 (\frac{x - 20}{3}) + 20$

Rewrite the expression.

$f (\frac{2 x}{3} - \frac{13}{3}) = x - 20 + 20$

Combine the opposite terms in $x - 20 + 20$ .

Add $- 20$ and $20$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = x + 0$

Add $x$ and $0$ .

$f (\frac{2 x}{3} - \frac{13}{3}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 x}{3} - \frac{13}{3}$ is the inverse of $f (x) = 3 (\frac{x - 9}{2}) + 20$ .

$f^{- 1} (x) = \frac{2 x}{3} - \frac{13}{3}$

Find the Inverse f(x)=3((x-9)/2)+20

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